Problem 32 - College Algebra

Question

For the following exercises, solve the equation involving absolute value. $$ |4 x+1|-3=6 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -\frac{5}{2}\).

Step 1: Isolate the Absolute Value Expression

The given equation is \(|4x + 1| - 3 = 6\). To isolate the absolute value, add 3 to both sides of the equation. \[ |4x + 1| = 6 + 3 \]Thus, the equation simplifies to:\[ |4x + 1| = 9 \]

Step 2: Set Up Two Separate Equations

Since we have \(|a| = b\), it implies that either \(a = b\) or \(a = -b\). Applying this to our equation, we set up two separate equations: 1. \(4x + 1 = 9\)2. \(4x + 1 = -9\)

Step 3: Solve the First Equation

Solve the equation \(4x + 1 = 9\) for \(x\):Subtract 1 from both sides:\[ 4x = 8 \]Divide both sides by 4:\[ x = 2 \]

Step 4: Solve the Second Equation

Solve the equation \(4x + 1 = -9\) for \(x\):Subtract 1 from both sides:\[ 4x = -10 \]Divide both sides by 4:\[ x = -\frac{10}{4} = -\frac{5}{2} \]

Step 5: Verify Solutions

To ensure the solutions are correct, substitute \(x = 2\) and \(x = -\frac{5}{2}\) back into the original equation.For \(x = 2\):\[ |4(2) + 1| - 3 = |9| - 3 = 9 - 3 = 6 \]For \(x = -\frac{5}{2}\):\[ |4(-\frac{5}{2}) + 1| - 3 = |-10 + 1| - 3 = |-9| - 3 = 9 - 3 = 6 \]Both solutions satisfy the original equation.

Key Concepts

Solving Absolute Value Equations

Solving equations that involve absolute value can be a straightforward process if you understand the basic properties of absolute value. The absolute value, denoted \(|a|\), of any number is its distance from zero on the number line, and it is always non-negative. When solving equations with absolute values, the formula \(|a| = b\) implies that there are actually two possibilities because the number could be positive or negative. Hence, \(|a| = b\) results in two equations: \(a = b\) and \(a = -b\).
Firstly, isolate the absolute value expression on one side. This allows you to apply the rule of two possible scenarios where you solve two different equations. By understanding these core principles, solving absolute value equations becomes much simpler.

Verification of Solutions

Verification of solutions is a crucial step to ensure that your answers are indeed correct. When dealing with absolute value equations, it's important to check both potential solutions against the original equation. This process helps verify their validity and ensures that there was no mistake in the algebraic calculations.

Here’s how to verify a solution:

• Substitute each solution back into the original equation.
• Simplify both sides of the equation to see if they equal.

By following this process, you will confirm whether the solutions are accurate. If they satisfy the original equation, they are valid. If not, then reevaluation is necessary. This reaffirms the importance of verification in maths to achieve reliable results.

Isolation of Absolute Value Expression

Isolation of the absolute value expression is the essential first step in solving absolute value equations. To isolate an absolute value expression, you need to manipulate the equation so that only the absolute value term is on one side.

For example, in the equation \( |4x + 1| - 3 = 6\), you have to remove other terms from around the absolute value before addressing it directly. Add 3 to both sides to get \( |4x + 1| = 9\).
By isolating the absolute value, you make it easier to apply the properties of absolute values which involve considering both the positive and negative scenarios. This step simplifies the problem and is critical for a successful solution.

Step-by-Step Solution Approach

A step-by-step solution approach is invaluable when tackling complex math problems, such as those involving absolute values. Following clear, logical steps ensures that you leave no room for errors and don’t overlook important details.

• Start with isolating the absolute value.
• Proceed to set up the two potential equations derived from the absolute value expression.
• Solve each equation separately to obtain possible solutions.
• Finally, verify the solutions by substituting back into the original equation.

This structured method not only paves the way for clearer problem-solving but also makes it easier for you to identify where the error might have occurred if your answers are incorrect. Adopting a step-by-step process reinforces understanding and ensures consistency in solving such problems.